Static optimization
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Contents |
Introduction
Static optimization is a method of musculoskeletal modeling that resolves resultant joint moments (typically calculated by inverse dynamics) into individual muscle forces. The need for an optimization procedure arises from the mechanical redundancy of the musculoskeletal system, where the number of unknown forces in structures spanning a joint exceeds the number of equations of dynamic equilibrium at the joint. This problem is often referred to as the force-sharing or general distribution problem (Crowninshield & Brand, 1981a; Caldwell & Chapman, 1991).
Static optimization introduces an additional equation (in the form of a cost function) that governs how the joint moments are distributed into any number of individual muscles. The "static" aspect of static optimization refers to the required assumption that muscle forces at one time step are independent of muscle forces at all other time steps (Hardt, 1978). In contrast, dynamic optimization solutions are time-dependent (Davy & Audu, 1987). Despite this independency, muscle force histories predicted by static optimization tend to be smooth and continuous, even if muscle activation and contraction dynamics are neglected.
A static optimization model requires as inputs the resultant joint moment histories and the muscle moment arms at these joints. If the moment arms are defined as functions of the joint positions, joint angle histories must be input as well. Additional information describing the muscles is usually included as well, in the form of physiological cross-sectional areas, maximum permissible forces, or Hill muscle model parameters. The model outputs the predicted muscle force histories during the movement in question. Static optimization has been widely used since the 1970s to predict muscle forces during human movement. See Crowninshield and Brand (1981a) and Erdemir et al. (2007) for reviews of a large number of static optimization studies.
Mathematical Basis
Cost Function
Static optimization-based estimates of muscle forces require the researcher to define a cost function that is minimized at each time step. This function will govern the muscle force distribution, and should be selected carefully. When muscle activation and contraction dynamics are neglected, the cost function at a given time step is typically the sum of weighted muscle forces raised to some power:
- <math>CF(t) = \sum_{m = 1}^M (a_m F_m)^n</math>
where <math>CF(t)</math> is the cost function score at time <math>t</math>, <math>M</math> is the number of muscles in the model, <math>F_m</math> is the force of muscle <math>m</math>, <math>a_m</math> is the weighting factor for muscle <math>m</math>, and <math>n</math> is a positive real number. The most widely-used cost function for static optimization of muscle forces during locomotion has been the sum of squared or cubed muscle stresses, where <math>a_m</math> = inverse of muscle physiological cross-sectional area and <math>n</math> = 2 or 3 (Crowninshield & Brand, 1981b). If the model considers muscle activation and contraction dynamics, the cost function is usually the sum of squared or cubed muscle activations (Kaufman et al., 1991; Anderson & Pandy, 2001; Yokozawa et al., 2007).
Constraints
The static optimization problem is constrained such that the muscle forces must produce the input resultant joint moments. This equality constraint takes the form of:
- <math>M_j(t) = r_j(t)F(t), j = 1,...,n_j</math>
where <math>M_j(t) is the resultant moment at joint <math>j</math> at time <math>t</math>, <math>r_j(t)</math> is the vector of muscle moment arms at joint <math>j</math> at time <math>t</math>, <math>F(t)</math> is the vector of muscle forces at time <math>t</math>, and <math>n_j</math> is the number of joints in the model. The joint moment equality constraint can also include ligament forces and bone contact forces if these forces are suspected to contribute substantially to the resultant joint moment. Values for <math>r_j(t)</math> are usually estimated from cadaver data or from musculoskeletal modeling software, and can be entered as time series or as functions of the joint angle trajectories.
Additionally, the muscle forces must be constrained with an inequality constraint to be greater than or equal to zero. Some models also place upper bounds on the muscle forces:
- <math>0 \leq F_m \leq F_{max}</math>
where <math>F_m</math> is the force in muscle <math>m</math> and <math>F_{max}</math> is the maximum permissible value of <math>F_m</math>.
Solution Derivation
Once the cost function and constraints are defined, the muscle forces can be calculated by solving the optimization problem. Linear and quadratic cost functions (<math>n \leq 2</math>) can be solved analytically using Lagrange multipliers, even for models with multiple degrees of freedom (Dul et al., 1984; Raikova & Prilutsky, 2001). Analytical solutions to nonlinear cost functions of higher orders (<math>n \geq 3</math>) cannot be derived for models with more than one degree of freedom because unique Lagrange multipliers cannot be calculated. For these types of models, numerical approximations for the global minimum of the cost function can be found using an optimization algorithm such as sequential quadratic programming, genetic evolution, or simulated annealing (Miller et al., in press).
Post-processing
Once the muscle force histories are known, it is common for researchers to use these forces to estimate the contact forces between bones at a joint. These forces are usually referred to as joint contact forces, bone contact forces, or articular contact forces in the literature. Assuming ligaments and other non-muscular forces are small, the contact forces can be found by balancing the force vectors of all the muscles spanning the joint and the bone contact force vector at the joint with the resultant joint force vector calculated by inverse dynamics:
- <math>\sum_{m = 1}^M \vec{F_m} + \vec{F_b} = \vec{F_{net}}</math>
where <math>\vec{F_m}</math> is the force vector of muscle <math>m</math> crossing the joint, <math>M</math> is the number of muscles spanning the joint, <math>\vec{F_b}</math>is the bone contact force vector at the joint, and <math>\vec{F_{net}}</math> is the resultant joint force vector. This calculation requires the lines of action of the muscle forces, which can be computed if the origins, insertions, and wrapping geometry of the muscles with respect to the bones are known. Researchers have used this method to estimate hip joint contact forces during walking (Crowninshield & Brand, 1981b) and ankle joint contact forces during running (Sasimontonkul et al., 2007).
If the muscle forces are calculated analytically or by a gradient-based optimization algorithm, it is unnecessary to check if the muscle forces satisfy the joint moment constraint. This constraint must be satisfied in order for the solution to exist. However, if a stochastic optimization algorithm such as simulated annealing is used, strict equality constraints can be impractical or impossible to enforce. In these cases, constraints are included as penalty functions that inflate the cost function score when constraints are violated. The algorithm is discouraged, but not restricted, from locating solutions that violate the constraints. To ensure that the algorithm has not settled on a solution with constraint violations, it is wise to multiply the output muscle forces by their moment arms, sum them across the appropriate joints to compute the predicted joint moments, and compare the predictions to the input joint moments.
History and Development
The earliest applications of static optimization to the general distribution problem are credited to Seireg and Arvikar (1973, 1975), who estimated lower extremity muscle forces during walking and squatting, and Penrod et al. (1974), who estimated upper extremity muscle forces. The authors used a linear cost function that minimized the sum of muscle forces. This method predicted unrealistic distributions of muscle forces (e.g. few active muscles, large forces in small muscles, maximum muscle forces during submaximal activities). The problems were due to the cost function, which distributes the joint moment into the muscle with the longest moment arm until that muscle force reaches its maximum, then distributes the moment into the muscle with the next-longest moment arm, etc., until the joint moment constraint is satisfied. The optimal solution for the minimization of muscle forces will thus always lie on a corner of the solution space.
Subsequent studies used different cost functions that were able to predict more realistic patterns of muscle forces. Crowninshield et al. (1978) estimated hip muscle forces during walking by minimizing the sum of muscle stresses. This cost function distributes the joint moment sequentially into the muscle with the largest product of moment arm and physiological cross-sectional area. The model predicted many muscles to be active at the same time, but only because the maximum muscle forces were limited to values expected during submaximal activity. The solution still lied on the corner of the solution space, with most muscles producing maximum allowed forces.
The linear cost functions of early static optimization studies were selected for mathematical convenience. At the time, many established computer algorithms for solving linear optimization problems were available, while few algorithms for solving nonlinear problems were in wide use. With developments in computational power and efficient nonlinear solvers, researchers began to use nonlinear cost functions. The advantage of the new nonlinear cost functions were that they predicted optimal solutions on the interior of the solution space; a muscle no longer had to reach its maximum force before another muscle was recruited to contribute to the joint moment. Crowninshield and Brand's (1981b) seminal paper on static optimization of muscle forces during walking proposed a cost function of the minimization of muscle stresses cubed, and proposed a phyiological rationale for the cost function based on the maximization of muscular endurance. The largest forces were predicted in the largest muscles (e.g. soleus, vasti, glutei) and the muscle force histories agreed well with electromyography patterns. This cost function is still in popular use today to predict muscle forces during human locomotion (Sasimontonkul et al., 2007; Edwards et al., in press).
Sensitivity of Solutions
A static optimization model requires a large number of input parameters. At a minimum, the muscle moment arms are required, and a decision on the degrees of freedom allowed at each joint must be made. Most models also include some parameters describing the architecture and force-production capabilities of the muscles. Sensitivity analyses have found that the predicted muscle forces can vary substantially with respect to these parameters.
Advantages and Criticisms
Compared to other methods for predicting muscle forces, static optimization is attractive for its speed. With efficient coding it is possible to find a static optimization solution in a small number of minutes on a desktop computer, even for a model with dozens of muscles. van den Bogert et al. (2008) presented a method for static optimization of muscle forces in nearly real-time as motion capture and force platform data are collected. Solutions to a dynamic optimization problem for a forward dynamics model with a comparable number of muscles can take hours, days, or even weeks to converge on an optimal solution, even on a very powerful computer. The computational time arises from the integration of the model's state equations, which must be performed many times to locate an optimal solution. These equations are not needed in static optimization problems. Anderson and Pandy (2001) found that static and dynamic optimization solutions for muscle forces during walking were nearly identical when the static optimization problem used the resultant joint moments from the dynamic optimization problem.
In general, it is not possible to measure individual muscle forces on human subjects, making it difficult to validate the forces predicted by static optimization. However, muscle force histories predicted by static optimization agree qualitatively with electromyography patterns from human subjects during walking and running (Crowninshield & Brand, 1981b; Glitsch & Baumann, 1997). The accuracy of static optimization has also been questioned in situations where in vivo forces were measured. Prilutsky et al. (1997) compared muscle forces of the ankle plantarflexors of a walking cat predicted by static optimization to those measured with tendon force transducers, and found that static optimization predicted muscle forces that were 43-119% larger than the measured forces, depending on the cost function used. However, the in vivo human Achilles tendon forces presented by Komi (1990) during running are comparable to the magnitude of plantarflexor force in a static optimization model of running (Miller et al., in press).
Static optimization has been criticized for lacking a strong physiological basis. The assumption that muscle forces are history-independent is a major deviation from the mechanics of in vivo muscle force production. Most models have not included the muscle activation and contraction dynamics that influence the muscle force history in vivo (Gasser & Hill, 1924; Hill, 1938; Gordon et al., 1965; Herzog, 2004). Including these dynamics appears to make little difference in the muscle force estimates for walking (Anderson & Pandy, 2001), but may be important for faster movements like running or sprinting. It is also unlikely that muscle force patterns during all human movements are governed by the proposed Crowninshield and Brand (1981b) maximum endurance criterion, since maximum endurance is not a major component of the motor goal for some movements (examples: sprinting, jumping for height, weight-lifting).
Cost functions based on the minimization of muscle stresses have been criticized for under-estimating the degree of muscular co-contraction (Gottlieb, 2000). Here co-contraction is defined as the presence of non-zero forces in muscles on both sides of a joint. To produce a given joint moment, any amount of co-contraction will result in greater muscle stresses compared to a solution with no co-contraction. The model will thus attempt to minimize co-contraction at all times. In fact, static optimization models will predict co-contraction only if the model includes bi-articular muscles. A model with only uni-articular muscles will never predict any co-contraction.
In summary, prediction of muscle forces by static optimization and the minimization of muscle stresses raised to the second or third power predicts muscle force histories that are likely reasonable estimates of muscle force patterns and relative magnitudes, at least on a qualitative basis. The exact magnitudes and temporal characteristics of the predicted forces are likely inaccurate to at least a small degree, and in some cases may be highly inaccurate. This accuracy is difficult to quantify since the true muscle forces are unknown in most cases. There are likely some situations where the stress minimization cost function is inappropriate due to inaccuracies of inverse dynamics estimates of joint moments (e.g. high-impact landings; Gruber et al., 1998) or maximum-effort activities where muscular endurance is not a large concern (e.g. vertical jumping, sprinting). In these situations, a forward dynamics model is likely a better choice for estimating muscle forces. An excellent discussion on the advantages and limitations of muscle force predictions by static optimization and muscle stress minimization is found in the target article by Prilutsky (2000) and the discussion articles that follow it (volume 4, issue 1 of Motor Control).
Studies using static optimization for movement analysis
The initial list is a summary from Erdemir et al., Model-based estimation of muscle forces exerted during movements. Clin. Biomech., 2006. List organized by activity than sorted by order of appearence.
Walking
A. Seireg and R.J. Arvikar, The prediction of muscular load sharing and joint forces in the lower extremities during walking, J. Biomech. 8 (1975), pp. 89–102.
R.D. Crowninshield, R.C. Johnston, J.G. Andrews and R.A. Brand, A biomechanical investigation of the human hip, J. Biomech. 11 (1978), pp. 75–85.
R.D. Crowninshield and R.A. Brand, A physiologically based criterion of muscle force prediction in locomotion, J. Biomech. 14 (1981), pp. 793–801.
A.G. Patriarco, R.W. Mann, S.R. Simon and J.M. Mansour, An evaluation of the approaches of optimization models in the prediction of muscle forces during human gait, J. Biomech. 14 (1981), pp. 513–525.
H. Rohrle, R. Scholten, C. Sigolotto, W. Sollbach and H. Kellner, Joint forces in the human pelvis–leg skeleton during walking, J. Biomech. 17 (1984), pp. 409–424.
R.A. Brand, D.R. Pedersen and J.A. Friederich, The sensitivity of muscle force predictions to changes in physiologic cross-sectional area, J. Biomech. 19 (1986), pp. 589–596.
J.J. Collins, The redundant nature of locomotor optimization laws, J. Biomech. 28 (1995), pp. 251–267.
U. Glitsch and W. Baumann, The three-dimensional determination of internal loads in the lower extremity, J. Biomech. 30 (1997), pp. 1123–1131.
D.R. Pedersen, R.A. Brand and D.T. Davy, Pelvic muscle and acetabular contact forces during gait, J. Biomech. 30 (1997), pp. 959–965.
Knee Flexion
J. Dul, M.A. Townsend, R. Shiavi and G.E. Johnson, Muscular synergism—I. On criteria for load sharing between synergistic muscles, J. Biomech. 17 (1984), pp. 663–673.
G. Li, K.R. Kaufman, E.Y. Chao and H.E. Rubash, Prediction of antagonistic muscle forces using inverse dynamic optimization during flexion/extension of the knee, J. Biomech. Eng. 121 (1999), pp. 316–322.
E. Forster, U. Simon, P. Augat and L. Claes, Extension of a state-of-the-art optimization criterion to predict co-contraction, J. Biomech. 37 (2004), pp. 577–581.
Running
U. Glitsch and W. Baumann, The three-dimensional determination of internal loads in the lower extremity, J. Biomech. 30 (1997), pp. 1123–1131.
Clenching
J.W. Osborn and F.A. Baragar, Predicted pattern of human muscle activity during clenching derived from a computer assisted model: symmetric vertical bite forces, J. Biomech. 18 (1985), pp. 599–612.
J.H. Koolstra, T.M. van Eijden, W.A. Weijs and M. Naeije, A three-dimensional mathematical model of the human masticatory system predicting maximum possible bite forces, J. Biomech. 21 (1988), pp. 563–576.
P.G. Trainor, K.R. McLachlan and W.D. McCall, Modelling of forces in the human masticatory system with optimization of the angulations of the joint loads, J. Biomech. 28 (1995), pp. 829–843.
Spinal Compression
A. Schultz, K. Haderspeck, D. Warwick and D. Portillo, Use of lumbar trunk muscles in isometric performance of mechanically complex standing tasks, J. Orthop. Res. 1 (1983), pp. 77–91.
A. Schultz, R. Cromwell, D. Warwick and G. Andersson, Lumbar trunk muscle use in standing isometric heavy exertions, J. Orthop. Res. 5 (1987), pp. 320–329.
J.C. Bean, D.B. Chaffin and A.B. Schultz, Biomechanical model calculation of muscle contraction forces: a double linear programming method, J. Biomech. 21 (1988), pp. 59–66.
V.K. Goel, W. Kong, J.S. Han, J.N. Weinstein and L.G. Gilbertson, A combined finite element and optimization investigation of lumbar spine mechanics with and without muscles, Spine 18 (1993), pp. 1531–1541.
R.E. Hughes, D.B. Chaffin, S.A. Lavender and G.B. Andersson, Evaluation of muscle force prediction models of the lumbar trunk using surface electromyography, J. Orthop. Res. 12 (1994), pp. 689–698.
R.E. Hughes, J.C. Bean and D.B. Chaffin, Evaluating the effect of co-contraction in optimization models, J. Biomech. 28 (1995), pp. 875–878.
R.E. Hughes and D.B. Chaffin, The effect of strict muscle stress limits on abdominal muscle force predictions for combined torsion and extension loadings, J. Biomech. 28 (1995), pp. 527–533.
M.A. Nussbaum, D.B. Chaffin and C.J. Rechtien, Muscle lines-of-action affect predicted forces in optimization-based spine muscle modeling, J. Biomech. 28 (1995), pp. 401–409.
M.A. Nussbaum and D.B. Chaffin, Pattern classification reveals intersubject group differences in lumbar muscle recruitment during static loading, Clin. Biomech. 12 (1997), pp. 97–106.
C.K. Cheng, H.H. Chen, C.S. Chen and S.J. Lee, Influences of walking speed change on the lumbosacral joint force distribution, Biomed. Mater. Eng. 8 (1998), pp. 155–165.
W.Z. Kong, V.K. Goel and L.G. Gilbertson, Prediction of biomechanical parameters in the lumbar spine during static sagittal plane lifting, J. Biomech. Eng. 120 (1998), pp. 273–280.
G.A. Hoek van Dijke, C.J. Snijders, R. Stoeckart and H.J. Stam, A biomechanical model on muscle forces in the transfer of spinal load to the pelvis and legs, J. Biomech. 32 (1999), pp. 927–933.
R.E. Hughes, Effect of optimization criterion on spinal force estimates during asymmetric lifting, J. Biomech. 33 (2000), pp. 225–229.
A. Shirazi-Adl, S. Sadouk, M. Parnianpour, D. Pop and M. El-Rich, Muscle force evaluation and the role of posture in human lumbar spine under compression, Eur. Spine J. 11 (2002), pp. 519–526.
F. Ezquerro, A. Simon, M. Prado and A. Perez, Combination of finite element modeling and optimization for the study of lumbar spine biomechanics considering the 3D thorax-pelvis orientation, Med. Eng. Phys. 26 (2004), pp. 11–22.
I.A. Stokes and M. Gardner-Morse, Muscle activation strategies and symmetry of spinal loading in the lumbar spine with scoliosis, Spine 29 (2004), pp. 2103–2107.
S.H. Brown and J.R. Potvin, Constraining spine stability levels in an optimization model leads to the prediction of trunk muscle cocontraction and improved spine compression force estimates, J. Biomech. 38 (2005), pp. 745–754.
A. Shirazi-Adl, M. El-Rich, D.G. Pop and M. Parnianpour, Spinal muscle forces, internal loads and stability in standing under various postures and loads–application of kinematics-based algorithm, Eur. Spine J. 14 (2005), pp. 381–392.
Neck Movements
S.P. Moroney, A.B. Schultz and J.A. Miller, Analysis and measurement of neck loads, J. Orthop. Res. 6 (1988), pp. 713–720.
Finger Movements
N. Brook, J. Mizrahi, M. Shoham and J. Dayan, A biomechanical model of index finger dynamics, Med. Eng. Phys. 17 (1995), pp. 54–63.
Wrist
D.D. Penrod, D.T. Davy and D.P. Singh, An optimization approach to tendon force analysis, J. Biomech. 7 (1974), pp. 123–129.
Elbow Movements
K.N. An, B.M. Kwak, E.Y. Chao and B.F. Morrey, Determination of muscle and joint forces: a new technique to solve the indeterminate problem, J. Biomech. Eng. 106 (1984), pp. 364–367.
R. Raikova, A general approach for modelling and mathematical investigation of the human upper limb, J. Biomech. 25 (1992), pp. 857–867.
J.H. Challis and D.G. Kerwin, An analytical examination of muscle force estimations using optimization techniques, Proc. Inst. Mech. Eng. 207 (1993), pp. 139–148.
R. Raikova, A model of the flexion–extension motion in the elbow joint some problems concerning muscle forces modelling and computation, J. Biomech. 29 (1996), pp. 763–772.
J.H. Challis, Producing physiologically realistic individual muscle force estimations by imposing constraints when using optimization techniques, Med. Eng. Phys. 19 (1997), pp. 253–261.
R. Raikova and H. Aladjov, The influence of the way the muscle force is modeled on the predicted results obtained by solving indeterminate problems for a fast elbow flexion, Comput. Methods Biomech. Biomed. Eng. 6 (2003), pp. 181–196.
R.T. Raikova, D.A. Gabriel and H.T.S. Aladjov, Experimental and modelling investigation of learning a fast elbow flexion in the horizontal plane, J. Biomech. 38 (2005), pp. 2070–2077.
J.E. Pierce and G. Li, Muscle forces predicted using optimization methods are coordinate system dependent, J. Biomech. 38 (2005), pp. 695–702.
Arm and Shoulder Movements
D. Karlsson and B. Peterson, Towards a model for force predictions in the human shoulder, J. Biomech. 25 (1992), pp. 189–199.
F.C. van der Helm, A finite element musculoskeletal model of the shoulder mechanism, J. Biomech. 27 (1994), pp. 551–569.
R. Happee, Inverse dynamic optimization including muscular dynamics, a new simulation method applied to goal directed movements, J. Biomech. 27 (1994), pp. 953–960.
R. Happee and F.C. van der Helm, The control of shoulder muscles during goal directed movements, an inverse dynamic analysis, J. Biomech. 28 (1995), pp. 1179–1191.
H. Nieminen, J. Niemi, E.P. Takala and E. Viikari-Juntura, Load-sharing patterns in the shoulder during isometric flexion tasks, J. Biomech. 28 (1995), pp. 555–566.
H. Nieminen, E.P. Takala, J. Niemi and E. Viikari-Juntura, Muscular synergy in the shoulder during a fatiguing static contraction, Clin. Biomech. 10 (1995), pp. 309–317.
J. Niemi, H. Nieminen, E.P. Takala and E. Viikari-Juntura, A static shoulder model based on a time-dependent criterion for load sharing between synergistic muscles, J. Biomech. 29 (1996), pp. 451–460.
T.S. Buchanan and D.A. Shreeve, An evaluation of optimization techniques for the prediction of muscle activation patterns during isometric tasks, J. Biomech. Eng. 118 (1996), pp. 565–574.
R.E. Hughes, M.G. Rock and K.N. An, Identification of optimal strategies for increasing whole arm strength using Karush–Kuhn–Tucker multipliers, Clin. Biomech. 14 (1999), pp. 628–634.
H.T. Lin, F.C. Su, H.W. Wu and K.N. An, Muscle forces analysis in the shoulder mechanism during wheelchair propulsion, Proc. Inst. Mech. Eng. 218 (2004), pp. 213–221.
S. van Drongelen, L.H. van der Woude, T.W. Janssen, E.L. Angenot, E.K. Chadwick and D.H. Veeger, Glenohumeral contact forces and muscle forces evaluated in wheelchair-related activities of daily living in able-bodied subjects versus subjects with paraplegia and tetraplegia, Arch. Phys. Med. Rehabil. 86 (2005), pp. 1434–1440.
References
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Dul J, Townsend MA, Shiavi R, and Johnson GE (1984). Muscular synergism - I. On criteria for load sharing between synergistic muscles. Journal of Biomechanics, 17, 663-673.
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Hardt DE (1978). Determining muscle forces in the leg during normal human walking - an application and evaluation of optimization methods. Journal of Biomechanical Engineering, 100, 72-78.
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Prilutsky BI (2000). Coordination of two- and one-joint muscles: functional consequences and implications for motor control. Motor Control, 4, 1-44.
Prilutsky BI, Herzog W, and Allinger T (1997). Forces of individual cat ankle extensor muscles during locomotion predicted using static optimization. Journal of Biomechanics, 30, 1025-1033.
Raikova RT and Prilutsky BI (2001). Sensitivity of predicted muscle forces to parameters of the optimization-based human leg model revealed by analytical and numerical analyses. Journal of Biomechanics, 34, 1243-1255.
van den Bogert AJ, Geijtenbeek T, and Even-Zohar O (2008). Real-time estimation of muscle forces from inverse dynamics. 4th North American Congress on Biomechanics, Ann Arbor, MI, USA, August 5-9, 2008.
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